Kinetic Space Plasma Turbulence

Transcript

1 Kinetic Space Plasma Turbulence PETER H. YOON 8the East-Asia School and Workshop on Laboratory, Space, and Astrophysical Plasmas July 30 (Mon) August 3 (Fri) 2018, Chungnam National University, Korea

2 Motivation Nonthermal charged particle distribution function in space Kappa f Max e v2 /vt 2 1 f kappa (1 + v 2 /κvt 2 )κ+1 Maxwell

3 Motivation Solar flare-generated radio bursts

4 Fundamentals of Plasma Kinetic Theory 1D, field-free electrostatic Vlasov-Poisson equation ( t + v x + e ) ae f a = 0, m a v E x = 4πˆn e a dv f a. a Separation into Average and Fluctuation f a (x, v, t) = F a (v, t) + δf a (x, v, t), E(x, t) = δe(x, t). Rewrite the equations ( t + e a δe ) ( F a + m a v t + v x + e a δe ) δf a = 0, m a v x δe = 4πˆn e a dv δf a. a

5 Random phase approximation < δf a >= 0, < δe >= 0. Stationary and homogeneous turbulence, < δf a (x, v, t) δf b (x, v, t ) >= g( x x, t t, v, v ).

6 ( t + e a δe ) ( F a + m a v t + v x + e a δe ) δf a = 0, m a v Upon averaging we have the formal particle kinetic equation F a t = e a m a v δf a δe. Inserting to the original equation, we obtain the equation for the perturbed distribution, ( t + v ) δf a = e a δe F a x m a v e a m a v (δf a δe < δf a δe >).

7 Fourier-Laplace transformation: δf a (x, v, t) = δf a kω(v, t) = δe(x, t) = δe kω (t) = dk 1 (2π) 2 dk 1 (2π) 2 L (slow time) dω δf a kω(v, t) e ikx iωt, dx 0 dt δf a (x, v, t) e ikx+iωt, dω δe kω (t) e ikx iωt, L dx dt δe(x, t) e ikx+iωt, where L = ( + iσ, + + iσ) (σ > 0 and σ 0: causality). 0

8 δe kω (t) = i 4πˆne a dv δf k kω(v, a t), a F a (v, t) = e a dk dω t m a v < δe k, ω(t) δfkω(v, a t) >, ( ω kv + i ) δf a t kω(v, t) = i e a δe kω (t) F a(v, t) m a v i e a dk dω [ δek ω m a v (t) δf k k a ω ω (v, t) < δe k ω (t) δf k k a,ω ω (v, t) >].

9 Absorb slow time into ω (short-cut trick), Short-hand notations: ω + i t ω. K = (k, ω), E K = δe kω, f K = δfkω, a F = F a, dk = dk dω, g K = i e a m a 1 ω kv + i0 v. Equation for the perturbed particle distribution: f K = g K F E K + dk g K (E K f K K < E K f K K >).

10 Iterative solution: f K = f (1) K + f (2) (n) K +, (f K En K). f (1) K = g K F E K, f (2) K = dk g K (E K f (1) K K < E K f (1) K K >) = dk g K g K K F (E K E K K < E K E K K >). More simplified notations: = dk 1 dk 2 δ(k 1 + K 2 K), 1+2=K E 1 = E K1, E 2 = E K2, g 1 = g K1, g 2 = g K2.

11 Iterative solution for f K f K = g K F E K + Symmetrized version f K = g K F E K + 1+2=K 1+2=K g K g 2 F (E 1 E 2 < E 1 E 2 >). 1 2 g 1+2 (g 1 + g 2 ) F (E 1 E 2 < E 1 E 2 >). Insert f K to Poisson equation, E K = i a 4πˆne a k dv f K.

12 Linear and nonlinear susceptibility response functions, ɛ(k) = 1 + i 4πe aˆn dv g K F, k a χ (2) a (1 2) = i 4πe aˆn dv g 1+2 (g 1 + g 2 ) F. 2 k 1 + k 2 or explicitly ɛ(k) = 1 + a χ (2) (1 2) = i 2 a [( v ωpa 2 F a / v dv k ω kv + i0, ωpa 2 1 dv k 1 + k 2 ω 1 + ω 2 (k 1 + k 2 ) v + i0 ) ] 1 ω 1 k 1 v + i0 + 1 Fa. ω 2 k 2 v + i0 v e a m a

13 0 = ɛ(k) E K + 1+2=K χ (2) (1 2) (E 1 E 2 < E 1 E 2 >). Multiply E K and take ensemble average 0 = ɛ(k) < E K E K > + χ (2) (1 2) < E 1 E 2 E K >. 1+2=K Homogeneous and stationary turbulence (and spectral representation), < E(x, t) E(x, t ) >=< E 2 > x x,t t, < E kω E k ω >= δ(k + k ) δ(ω + ω ) < E 2 > kω. 0 = ɛ(k) < E 2 > K + χ (2) (1 2) < E 1 E 2 E K >. 1+2=K

14 Closure of Hierarchy If E K is linear eigenmode, then ɛ(k) E K = 0, and < E 1 E 2 E K >= 0, but for nonlinear system we write E K = E (0) K + E(1) K, where ɛ(k) E(0) K = 0. Then E (1) K 1 ( ) dk χ (2) (K K K ) E (0) K ɛ(k) E(0) K K < E(0) K E(0) K K >. Three-body correlation, < E K E K K E K > < E (0) K E (0) K K E (0) K }{{ > } + < E(1) 0 K E (0) K K E (0) K > + < E (0) K E (1) K K E (0) K > + < E(0) K E (0) K K E (1) K >.

15 < E K E K K E K >= 1 dk χ (2) (K K K ) ɛ(k ) ( E K E K K E K K E K E K E K K E K K E K ) 1 dk χ (2) (K K K K ) ɛ(k K ) ( E K E K K K E K E K E K E K K K E K E K ) 1 dk χ (2) ( K K + K ) ɛ( K) ( E K E K K E K E K+K E K E K K E K E K+K ). where we dropped the superscript (0). To obtain < E 3 > one needs < E 4 >.

16 For homogeneous and stationary turbulence, < E K E K E K E K >= δ(k + K + K + K ) Useful symmetry relations: [ δ(k + K ) < E 2 > K < E 2 > K +δ(k + K ) < E 2 > K < E 2 > K + δ(k + K ) < E 2 > K < E 2 > K ] + < E 4 > K;K+K ;K+K +K. χ (2) ( 1 2) = χ (2) (1 2), χ (2) (1 2) = χ (2) (2 1), χ (2) (1 2) = χ (2) (2 1) = χ (2) ( ).

17 Three-body cumulants: < E K E K K E K > = 2χ(2) (K K K ) ɛ(k ) + 2χ(2) (K K K ) ɛ(k K ) 2χ(2) (K K K ) ɛ (K) < E 2 > K K < E 2 > K < E 2 > K < E 2 > K < E 2 > K < E 2 > K K.

18 Spectral balance equation 0 = ɛ(k) < E 2 > K ( {χ +2 dk (2) (K K K )} 2 ɛ(k < E 2 > K K < E 2 > K ) + {χ(2) (K K K )} 2 ɛ(k K ) χ(2) (K K K ) 2 ɛ (K) < E 2 > K < E 2 > K < E 2 > K < E 2 > K K ).

19 Formal Wave Kinetic Equation Reintroduce the slow time dependence, ω ω + i t. This leads to ( ɛ(k, ω) < E 2 > kω ɛ k, ω + i ) < E 2 > kω t ( ɛ(k, ω) + i ) ɛ(k, ω) < E 2 > kω. 2 ω t

20 0 = i ɛ(k, ω) 2 ω t < E2 > kω +ɛ(k, ω) < E 2 > kω ( + 2 dk dω {χ (2) (k, ω k k, ω ω )} 2 ɛ(k, ω ) E 2 k k,ω ω E 2 kω + {χ(2) (k, ω k k, ω ω )} 2 ɛ(k k, ω ω ) χ(2) (k, ω k k, ω ω ) 2 ɛ (k, ω) E 2 k ω E 2 kω E 2 k ω E 2 ) k k,ω ω Real part leads to the dispersion relation, and imaginary part leads to the wave kinetic equation..

21 Particle kinetic equation For particles, leading order perturbed distribution is sufficient, F a = e a dk t m a v < E K fk a >, where F a t = Re i e2 a m 2 a f K = f (1) K + f (2) K +. dk dω v < E2 > kω F a ω kv + i0 v.

22 Particle Kinetic Equation F a t = Re i e2 a m 2 a v dk dω < E2 > kω F a ω kv + i0 v. Dispersion Relation Re ɛ(k, ω) < E 2 > kω = 0. For dispersion relation nonlinear terms are ignored.

23 Wave Kinetic Equation t < 2 Im ɛ(k, ω) E2 > kω = Re ɛ(k, ω)/ ω < E2 > kω 4 Re ɛ(k, ω)/ ω Im dk dω [ ( < E {χ (2) (k, ω k k, ω ω )} 2 2 > k k,ω ω ɛ(k, ω ) + < ) E2 > k ω ɛ(k k, ω ω < E 2 > kω ) χ(2) (k, ω k k, ω ω ) 2 ] ɛ < E 2 > k (k, ω) ω < E2 > k k,ω ω.

24 Linear and Nonlinear Suceptibilities χ a (k, ω) = ω2 pa k χ (2) a dv F a / v ω kv + i0, (k 1, ω 1 k 2, ω 2 ) = i e a ω 2 pa 2 m a k 1 + k 2 1 ω 1 + ω 2 (k 1 + k 2 )v + i0 v ( Fa / v ω 2 k 2 v + i0 + F a / v ω 1 k 1 v + i0 dv ).

25 Partial integrations, χ a (k, ω) = ω 2 pa dv F a (ω kv + i0) 2. χ (2) a (k 1, ω 1 k 2, ω 2 ) = k 1 k 2 (k 1 + k 2 ) dv F a (ω 1 k 1 v)(ω 2 k 2 v) ( 1 k 1 ω 1 + ω 2 (k 1 + k 2 )v ω 1 k 1 v ), + k 2 ω 2 k 2 v + k 1 + k 2 ω 1 + ω 2 (k 1 + k 2 )v where the small positive imaginary parts +i0 are implicit in the resonance denominators.

26 Symmetry Relations χ (2) (k 2, ω 2 k 1, ω 1 ) = χ (2) (k 1, ω 1 k 2, ω 2 ). ɛ( k, ω) = ɛ (k, ω), χ (2) ( k 1, ω 1 k 2, ω 2 ) = χ (2) (k 1, ω 1 k 2, ω 2 ). χ (2) (k 1, ω 1 k 2, ω 2 ) = χ (2) (k 2, ω 2 k 1, ω 1 ) = χ (2) (k 1 + k 2, ω 1 + ω 2 k 2, ω 2 ).

27 Linear Dielectric Susceptibility For fast wave ω kv a th, χ a (0, ω) = ω 2 pa/ω 2. ( ) χ a (k, ω) = ω2 pa ω k2 T a m a ω 2 iπ ω2 pa k dv F a v δ(ω kv). For slow wave ω kv a th χ a (k, ω) = 2ω2 pa k 2 vth a2 iπ ω2 pa k dv F a v δ(ω kv).

28 Nonlinear Susceptibility χ (2) a (0, ω 1 0, ω 2 ) = 0, χ (2) a (k 1, 0 k 2, 0) = ie a T a Fast-Wave Approximation ω 2 pa k 1 k 2 k 1 + k 2 ω 1 k 1 v a th, ω 2 k 2 v a th, ω 1 + ω 2 k 1 + k 2 v a th, 1 v a2 th. χ (2) a (k 1, ω 1 k 2, ω 2 ) = i 2 ( k1 e a m a ω 1 + k 2 ω 2 + k 1 + k 2 ω 1 + ω 2 ω 2 pa ω 1 ω 2 (ω 1 + ω 2 ) ).

29 Two Fast Waves and a Slow Wave ω 1 is arbitrary and ω 2 k 2 v a th, ω 1 + ω 2 k 1 + k 2 v a th, a (k 1, ω 1 k 2, ω 2 ) = i e a ωpa 2 2 m a ω 2 (ω 1 + ω 2 ) [k 2 dv F a/ v ω 1 k 1 v ( k2 + k ) ] 1 + k 2 F a ω 2 ω 1 + ω dv 2 ω 1 k 1 v, χ (2) where we have used the identity F a dv (ω 1 k 1 v) 2 = 1 k 1 dv F a/ v ω 1 k 1 v.

30 χ (2) e a a (k 1, ω 1 k 2, ω 2 ) = i k 1 2 m a ω 2 (ω 1 + ω 2 ) χ a(k 1, ω 1 ) e a ω 2 pa i 2 m a ω 2 (ω 1 + ω 2 ) ( 2 k 1 vth a2 iπ dv F ) a v δ(ω 1 k 1 v).

31 Re ɛ(k, ω) < E 2 > kω = 0, F a = Re i e2 a t m 2 dk dω < E2 > kω F a a v ω kv + i0 v, t < 2 Im ɛ(k, ω) E2 > kω = Re ɛ(k, ω)/ ω < E2 > kω 4 Re ɛ(k, ω)/ ω Im dk dω [ ( < E {χ (2) (k, ω k k, ω ω )} 2 2 > k k,ω ω ɛ(k, ω ) + < ) E2 > k ω ɛ(k k, ω ω < E 2 > kω ) ] χ(2) (k, ω k k, ω ω ) 2 ɛ < E 2 > k (k, ω) ω < E2 > k k,ω ω.

32 Re ɛ(k, ω) < E 2 > kω = 0, ω = ωk α (α = L, S). < E 2 > kω = [ I +α k δ(ω ωk α ) + I α k δ(ω + ωk α ) ] α = Ik σα δ(ω σωk α ). σ=±1 α

33 Langmuir (α = L) and ion-sound wave (α = S) ωk L = ω pe (1 + 3 ) ( 2 k2 λ 2 De = ω pe k 2 v 2 ) T e 4 ωpe 2, ω k L = ωk L, ω S k = kc S (1 + 3T i /T e ) 1/2 (1 + k 2 λ 2 De )1/2, ω S k = ω S k. Particle kinetic equation, F a t D = πe2 a m 2 a ( D F a v = v σ=±1 α ), dk Ik σα δ(σωα k kv).

34 < E 2 > kω = σ=±1 α=l,s < E 2 > k ω = σ =±1 β=l,s < E 2 > k k,ω ω = I σα k δ(ω σω α k ), σ =±1 γ=l,s I σ β k δ(ω σ ω β k ), I σ γ k k δ(ω ω σ ω γ k k ). 1 ɛ(k, ω) = P 1 ɛ(k, ω) α=l,s σ=±1 1 ɛ (k, ω) = P 1 ɛ (k, ω) + α=l,s σ=±1 iπ δ(ω σωk α) Re ɛ(k, σωk α, )/ σωα k iπ δ(ω σωk α) Re ɛ(k, σωk α. )/ σωα k

35 Ik σα 2 Im ɛ(k, σωk α ) = t Re ɛ(k, σωk α)/ σωα k 4 Im dk Re ɛ(k, σωk α)/ σωα k + σ =±1 σ =±1 β γ I σα k P {χ(2) (k, σω α k σ ω γ k k k k, σ ω γ k k )} 2 ɛ(k, σω α k σ ω γ k k ) P {χ(2) (k, σ ω β k k k, σω α k σ ω β k )} 2 ɛ(k k, σω α k σ ω β k ) I σ β k ] Ik σα I σ γ k k I σα k

36 4π + Re ɛ(k, σωk α)/ σωα k σ,σ =±1 β,γ [ Im dk {χ (2) (k, σ ω β k k k, σ ω γ k k )} 2 ( I σ γ k k I σα k Re ɛ(k, σ ω β k )/ σ ω β + k + χ(2) (k, σ ω β k k k, σ ω γ k k ) 2 Re ɛ(k, σω α k )/ σωα k δ(σω α k σ ω β k σ ω γ k k ). I σ β k I σα k Re ɛ(k k, σ ω γ k k )/ σ ω γ k k ] I σ β k I σ γ k k )

37 Ik σl = 2 Im ɛ(k, σωl k ) t ɛ (k, σωk L) 4 ɛ (k, σωk L) Im 8π + ɛ (k, σωk L) σ =±1 σ,σ =±1 I σl k ( I σ L k Iσ S S k k ɛ (k, σωk L) Iσ k k IσL k ɛ (k, σ ω L k ) δ(σω L k σ ω L k σ ω S k k ) ], [ ] dk A σ,σ L,L (k, k ) I σ L k + Aσ,σ L,S (k, k ) I σ S k Ik σl dk [ χ (2) (k, σ ω L k k k, σω L k σ ω L k ) 2 I σ L k IσL k ɛ (k k, σ ω S k k ) )

38 I σs k t = 2 Im ɛ(k, σωs k ) ɛ (k, σωk S) 4 ɛ (k, σωk S) Im + 4π ɛ (k, σω S k ) σ =±1 σ,σ =±1 I σs k ( I σ L k Iσ L L k k ɛ (k, σωk S) Iσ k k IσS k ɛ (k, σ ω L k ) δ(σω S k σ ω L k σ ω L k k ) ], [ ] dk A σ,σ S,L (k, k ) I σ L k + Aσ,σ S,S (k, k ) I σ S k Ik σs dk [ χ (2) (k, σ ω L k k k, σω S k σ ω L k ) 2 I σ L k IσS k ɛ (k k, σ ω L k k ) ) A σ,σ α,β (k, k ) = 2 {χ (2) (k, σ ω β k k k, σω α k σ ω β k )} 2 P 1 ɛ(k k, σω α k σ ω β k ).

39 See the following references for details: P. H. Yoon, Generalized weak turbulence theory, POP, 7, 4858 (2000) L. F. Ziebell, R. Gaelzer, and P. H. Yoon, Nonlinear development of weak beam-plasma instability, POP, 8, 3982 (2001) P. H. Yoon, Effects of spontaneous fluctuations on the generalized weak turbulence theory, POP, 12, (2005) P. H. Yoon, T. Rhee, and C.-M. Ryu, Effects of spontaneous thermal fluctuations on nonlinear beam-plasma interaction, POP, 12, (2005)

40 Induced Emission: Linear Wave-Particle Resonance Ik σl t = π σωk L ind. emiss. Ik σs t = π µ k σωk L ind. emiss. µ k µ k = k 3 λ 3 De ω 2 pe k ω 2 pe k ( F e + me me dv δ(σω L k kv) Fe F i m i m i k, v IσL dv δ(σω S k kv) v ) I σs k µ k, 1 + 3Ti T e.

41 Decay/Coalescence: Nonlinear Three-Wave Resonance I σl k t [ = π decay 2 σω L k I σ L k e 2 σω L Te 2 k σ,σ =±1 I σ S k k µ k k ( σ ω L k δ(σω L k σ ω L k σ ω S k k ), dk µ k k (k k ) 2 I σ ) S k k + σ ω L L k k µ Iσ k Ik σl k k Ik σs t = π e 2 σω S decay µ k 4 Te 2 k dk µ k k 2 σ,σ =±1 [ ( σωk L I σ L L k Iσ k k σ ω L L k Iσ k k + σ ω L L k k Iσ k δ(σω S k σ ω L k σ ω L k k ). ) ] I σs k µ k ]

42 Induced Scattering: Nonlinear Wave-Particle Resonance I σl k t = σωk L scatt. π ω 2 pe e 2 m e m i σ =±1 dk (k k ) F i v δ[σωl k σ ωk L (k k ) v] Ik σ dv L IσL k.

43 Adding effects of spontaneous thermal fluctuation This tutorial is too short to discuss in detail but in general, induced processes must be balanced by spontaneous processes. F e t A i = = ( ) F e A i F e + D ij, v i v j e 2 dk k i 4πm e k 2 σωk L δ(σωk L k v), D ij = πe2 m 2 e dk k i k j k 2 σ=±1 σ=±1 δ(σω L k k v) I σl k.

44 Langmuir Wave Kinetic Equation I σl k t + 2 = πω2 pe k 2 σ,σ =±1 σω L k ˆn e2 dv δ(σωk L k v) π Fe }{{} dk π e 2 4 Te 2 spont. emission µ k k (k k ) 2 k 2 k 2 k k 2 + σωk L Ik σl }{{} δ(σωk L σ ω L k σ ω S k k ( ) ) σωk L I σ L S k Iσ k k }{{} σ ω L S k Iσ k k IσL k σ ω L L k k Iσ k IσL k }{{} spont. decay induced decay k Fe v induced emission

45 πe2 m 2 e ωpe 2 σ =±1 dk dv (k k ) 2 k 2 k 2 δ[σω L k σ ω L k (k k ) v] ˆn e2 σω L ( πωpe 2 k σ ω L k IσL k σωk L I σ L) k (Fe + F i) }{{} spontaneous scattering ( ) + I σ L k IσL k (k k ) (σωk L σ ω L me k ) Fe (σωk L ) F i. v m i }{{} induced scattering

46 Forward/backward-Ion-sound Wave Kinetic Equation I σs k t = πµ kω 2 pe k 2 ˆn e2 dv δ(σω S k k v) ( k (Fe + Fi) + σωk L Ik σs F i } π {{} v m i }{{} spont. emission σ,σ =±1 σω L k induced emission )( F e + me ) dk π e 2 µ k [k (k k )] 2 δ(σω S 4 Te 2 k 2 k 2 k k 2 k σ ω L k σ ω L k k ) ) ( σωk L I σ L L k Iσ k k }{{} σ ω L L k Iσ k k IσS k σ ω L L k k Iσ k IσS k }{{} spont. decay induced decay.

47 Beam-Plasma Instability and Langmuir Turbulence Quasi-stationary ions F i = e v2 /v 2 T i π 1/2 v T i. Initial beam plus background electron distribution F e (v, 0) = (1 δ) e v2 /v 2 T e π 1/2 v T e + δ e (v v0)2/v2 T e π 1/2 v T e.

48 Beam-Plasma Instability and Langmuir Turbulence

54 Beam-Plasma Instability and Langmuir Turbulence f kappa 1 (1 + v 2 /κvt 2, κ = 2.25 )κ+1

55 Quiet Time Solar Wind Electrons f kappa 1 (1 + v 2 /κv 2 T )κ+1, FAST WIND e L SUN SLOW WIND EARTH As solar wind expands, f e approaches f κ with κ approaching κ 2.25.

56 Turbulent Equilibrium and Kappa Distribution

57 Turbulent Equilibrium and Kappa Distribution [ v 2 ] κ 1 f e (v) 1 + ( ) κ 3, 2 v 2 T e ( ) I(k) = T e κ 3 ω 2 pe 2 4π ( ) κ + 1 κ 3. 2 (kvt e ) 2 T e T i κ 3/2 κ + 1, and κ = 9 4 = 2.25.

58 Quiet Time Solar Wind Electron Distribution f(v) (s 3 m -6 ) f(v) (s 3 m -6 ) STEREO B ~v ~v v (m/s) Jan 9 T=9.81 ev (Core) 2007 Nov 30 STEREO A STEREO A ~v WIND EESA-L STEREO B ~v v (m/s) WIND EESA-H Kappa=7.02 (Halo) Theory: f e (v) 1 v 2κ+2 ~ v 6.5. Observa/on: (Super Halo) f e (v) v 5.5 v 7.5

59 Conclusion QUiet-time solar wind electrons feature inverse power-law tail population f(v) v α, where α is close to 6.5. This is explained by kappa distribution with index κ = 2.25, which corresponds to an asymptotic state of Langmuir turbulence. Kinetic plasma turbulence theory successfully explains a long standing problem of the physical origin of nonthermal electron velocity distribution.